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Adsorption and self-assembly of linear polymers on surfaces: acomputer simulation study

Citation for published version:Chremos, A, Glynos, E, Koutsos, V & Camp, PJ 2009, 'Adsorption and self-assembly of linear polymers onsurfaces: a computer simulation study', Soft Matter, vol. 5, no. 3, pp. 637-645.https://doi.org/10.1039/b812234b

Digital Object Identifier (DOI):10.1039/b812234b

Link:Link to publication record in Edinburgh Research Explorer

Document Version:Peer reviewed version

Published In:Soft Matter

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Download date: 01. Feb. 2021

Adsorption and self-assembly of linear polymers on surfaces: a

computer simulation study**

A. Chremos,1 E. Glynos,

2,† V. Koutsos

2 and P.J. Camp

1,*

[1]EaStCHEM, School of Chemistry, Joseph Black Building, University of Edinburgh, West Mains

Road, Edinburgh, EH9 3JJ, UK.

[2]Institute for Materials and Processes, School of Engineering and Electronics, and Centre for

Materials Science and Engineering, The University of Edinburgh, Mayfield Road, Edinburgh, EH9

3JL, UK.

[†

]Present address: Department of Materials Science and Engineering, University of Michigan, Ann

Arbor, MI 48109-2136, USA.

[*

]Corresponding author; e-mail: philip.camp@ed.ac.uk

[**

]The authors are grateful to Jacques Roovers for the synthesis of the PB polymers. This research

was supported by the School of Chemistry at the University of Edinburgh through the award of a

studentship to A. C.; and by the EPSRC DTA, and the Institute for Materials and Processes in the

School of the Engineering and Electronics at the University of Edinburgh, through the award of a

studentship to E. G.

Graphical abstract:

Summary:

Computer simulations are used to gain insight on polymer adsorption and self assembly on smooth

surfaces, as studied in recent atomic force microscopy experiments.

Post-print of peer-reviewed article published by the Royal Society of Chemistry.

Published article available at: http://dx.doi.org/10.1039/B812234B

Cite as:

Chremos, A., Glynos, E., Koutsos, V., & Camp, P. J. (2009). Adsorption and self-assembly

of linear polymers on surfaces: a computer simulation study. Soft Matter, 5(3), 637-645.

Manuscript received: 17/07/2008; Accepted: 15/10/2008; Article published: 21/11/2008

Page 1 of 23

Abstract

The adsorption and self-assembly of linear polymers on smooth surfaces are studied using coarse-

grained, bead-spring molecular models and Langevin dynamics computer simulations. The aim is to

gain insight on atomic-force microscopy images of polymer films on mica surfaces, adsorbed from

dilute solution following a good solvent-to-bad solvent quenching procedure. Under certain

experimental conditions, a bimodal cluster distribution is observed. It is demonstrated that this type of

distribution can be reproduced in the simulations, and rationalized on the basis of the polymer

structures prior to the quench, i.e., while in good-solvent conditions. Other types of cluster

distribution are described and explained. Measurements of the fraction of monomers bound to the

surface, the film height, and the radius of gyration of an adsorbed polymer chain are also presented,

and the trends in these properties are rationalized. In addition to providing insight on experimental

observations, the simulation results support a number of predicted scaling laws such as the decay of

the monomer density as a function of distance from the surface, and scaling of the film height with the

strength of the polymer -surface interactions.

1. Introduction

Polymers near to, or adsorbed on, surfaces exhibit useful and interesting properties. Adsorbed

polymers find application in colloid stabilization,1,2

nanoscale surface patterning ,3 friction

modification,4,5

DNA microarrays6 and adhesion.

7 Polymers can be attached to appropriate surfaces

either through chemisorption/grafting (i.e., anchoring by chemical bonds) or by physisorption (i.e.,

chain attachment by van der Waals interactions). For a weakly adsorbing surface the physisorption

and resulting conformational relaxation of the chain is driven by the competition between the entropic

repulsion due to the loss of conformational freedom and the drop in energy from binding monomers to

the substrate. Earlier investigations have focused on thin-film polymer blends,8–10

block copolymer

micelles adsorbed on surfaces,11–13

end-grafted polymers chemisorbed on surfaces,14–16

and several

other complex systems. The simple case of homopolymer chains physisorbed on a substrate has been

studied with simulations of free17–23

and tethered chains,24–26

and through theoretical approaches;27–29

experimental studies, however, are scarce.

In recent work by our groups , we studied the physisorption and self-assembly of star30

and linear31

polymers on smooth surfaces using atomic-force microscopy (AFM ). In a typical experiment, a

polymer solution was prepared in a good solvent at concentrations below the critical overlap volume

fraction ( *), resulting in well-separated chains in solution and hence precluding any strong degree of

structural ordering within the polymer component. Polymer (sub-)monolayers were formed by

exposing a smooth surface (such as highly ordered pyrolytic graphite or mica) to the polymer

Page 2 of 23

solution. The surface was then placed in a good solvent bath for several hours and extensively rinsed

with good solvent to remove any non-adsorbed polymer . Finally, the samples were dried gently under

a stream of nitrogen and subsequently imaged in air by AFM in tapping mode to investigate the

resulting structures from this good solvent-to-bad solvent ‘quench’. Depending on the polymer

molecular weight, architecture, and concentration, very different surface structures can be obtained.

For the case of star polymers it was found that the functionality (number of arms) and concentration

of star polymers controls a crossover between ‘polymer ’ and ‘soft-colloid’ regimes, being

distinguished by characteristic cluster topologies, sizes, and surface coverages.30

Using the same

experimental procedure, we have now studied the adsorption of linear polybutadiene (PB) on to mica

from toluene. A report of this investigation is in preparation,31

but for the purposes of the current

work, we present one key experimental observation on which we will aim to gain insight using

computer simulations. In Fig. 1(a) we present an AFM image of a freshly cleaved mica surface

exposed for 30 minutes to a toluene PB solution (molecular weight 78.8 kg mol−1

) leading to a surface

coverage of 7.75 × 10−2

mg m−2

. This image shows two distinct types of cluster . In Fig. 1(b) we show

the corresponding bimodal cluster -height distribution, with most probable heights of approximately 1

nm and 6 nm. In this work we aim to reproduce, and gain insight on, the development of bimodal

cluster distributions using Langevin dynamics simulations of coarse-grained ‘bead-spring’ models of

linear polymers .

Figure 1. (a) AFM image of linear polybutadiene (molecular mass 78.8 kg mol−1

) adsorbed on mica

from toluene at a surface concentration of 7.75 × 10−2

mg m−2

. The image size is 6 × 6 µm2. (b)

Cluster -height distribution corresponding to the AFM image in (a).

In the experimental quenching procedure described above, the microscopic structures of the polymer

solutions in contact with surfaces prior to quenching must control the nature of polymer adsorption.

Hence, the behaviour of polymers in good-solvent conditions and near a surface is of considerable

Page 3 of 23

interest. In the past, adsorption and depletion-layer effects have been the subjects of many

experimental, theoretical, and computer-simulation studies. Of particular relevance to the current

work is the seminal study by de Gennes focused on the monomer volume-fraction profile (z) as a

function of the perpendicular distance from the surface z.32

De Gennes considered a semidilute

solution of chains in contact with a weakly attracting wall, with the wall-monomer interaction of a

range comparable to the monomer size a. In the semidilute regime, the polymer volume fraction in

solution b > *, meaning that there are overlapping chains. Near the wall, chains are physisorbed

through a small number of monomers, leading to the formation of loops with characteristic dimension

D > a. Well away from the wall, the characteristic length scale is the bulk correlation length ξb

b−3/4

,27

which in the semidilute regime is comparable to the polymer radius of gyration RG in bulk

solution. Three regimes of z can be identified: the proximal regime z a<D where (z) is dictated by

the short-range interactions with the wall; the central regime D < z < ξb in which ‘no other length

enters in the problem’32

meaning that if ξ[ (z)] z and ξ(z) (z)−3/4

then (z) z−4/3

; and the distal

regime z >ξb where [ (z) − b]/ b exp(– z/ξb). The structure in the central regime is ‘self-similar’ or

‘scale-free’, because there is no characteristic length scale (unlike in the proximal or distal regimes,

which are characterized by a and ξb, respectively). The existence of a self-similar structure

characterized by an exponent of −4/3 has been confirmed experimentally by neutron scattering 33

and

by neutron reflectivity .34–36

Results from Monte Carlo simulations of lattice polymer models are also

consistent with this behaviour.37–39

An incidental result of the current work is a demonstration that a

coarse-grained, off-lattice polymer model can reproduce this self-similar structure, and with the

correct exponent.

In this paper we report a simulation study of adsorbed linear-polymer films. We use Langevin

dynamics simulations of coarse-grained ‘bead-spring’ models to gain insight on the results from

polymer adsorption experiments. The outline of the study is as follows. Firstly, we study the

properties of isolated adsorbed polymers (vanishing surface coverage). This situation has been

considered many times before (see, e.g., Ref. 23), but for the purposes of comparison with the case of

finite surface coverage, we reconsider specific single-molecule structural properties for the particular

coarse-grained models being employed. Next, we deal with many polymers on a surface under good

solvent conditions, corresponding to the prelude of the bad-solvent quench in experiments. Of

particular interest are simulation measurements of (z) and the comparison with de Gennes'

theoretical predictions. Finally, we simulate the good solvent-to-bad solvent quench, and its effects on

the structure of the polymerfilm. Specifically, we identify under what conditions a bimodal cluster

distribution (such as those seen in experiments—Fig. 1) should be expected. The paper is organized as

follows. Section II contains details of the coarse-grained polymer model, and the simulation methods.

Results for isolated polymers and many polymers are presented in Section III. Section IV concludes

the paper.

Page 4 of 23

Simulation model and methods

Linear polymers are modelled as chains of N coarse-grained ‘beads’ connected by ‘springs’. The

physical motivation for such a model stems from the fact that correlations between monomers die off

beyond a characteristic length, called the Kuhn length b. Hence, if a number of contiguous monomers

are rendered by a single bead of dimension b, then the scaling properties of the chain on length scales

larger than b will be left invariant.27,40,41

Such bead-spring models of linear and star polymers were

first introduced and employed in simulations by Grest and co-workers.42–45

In this work, Nbeads of

equal mass m are connected to form a chain using a non-linear finitely extensible (FENE) potential

between neighbouring beads, given by

(1)

Here r is the bead-bead separation, R0 is the maximum possible (bonded) bead-bead separation, and k

is a spring constant. In this study we use parameters from earlier work,43

namely R0 = 3σ/2 and k =

30ε/σ2; ε and σ are the energy and range parameters, respectively, for the non-bonded interactions to

be defined next.

The non-bonded interactions operate between all pairs of beads, and are derived from a composite

potential devised by Steinhauser.46

The potential is a combination of three terms. First we write the

purely repulsive Weeks-Chandler-Andersen (WCA) potential,47

which is a Lennard-Jones potential

cut and shifted at the position of the minimum, rmin = 21/6σ:

(2)

To represent the attractive interactions, the WCA potential is shifted back down in the range 0 ≤ r ≤

rmin by a square-well (SW) potential

(3)

where λ reflects the solvent quality (to be discussed below). To interpolate the potential smoothly

between −λε at r = rmin and 0 at a cut-off distance rcut > rmin, we add the term

Page 5 of 23

(4)

α and β satisfy the conditions αrmin2 + β = π and αrcut

2 + β = 2π. The cosine form of the potential also

means that dVcos/dr = 0 at r = rcut. Following earlier work,46

we choose rcut = 3σ/2, for which the

appropriate parameters are

(5)

(6)

The final, non-bonded potential is V(r) = VWCA(r) + VSW(r) + Vcos(r), and is sketched in Fig. 2. The

parameter λ controls the depth of the potential well at r = rmin, and provides a convenient measure of

the solvent quality. In a good solvent , the effective bead-bead interactions are purely repulsive; this

corresponds to λ = 0. In a bad solvent , the bead-bead interactions are attractive, and this behaviour

can be modelled with λ = 1; this corresponds to an attractive well of depth ε which sets the molecular

energy scale. θ-solvent conditions—under which the chain statistics are very similar to those for an

ideal (non-interacting) chain—are reproduced by λ = 0.646.46

Figure 2. The non-bonded, bead-bead interaction potential V(r) with λ = 0.7, showing the

contributions from VWCA(r) + VSW(r) (blue) and Vcos(r) (red).

Page 6 of 23

For simulations involving a surface, an additional effective bead-surface potential is used,45

based on

integrating the Lennard-Jones interactions arising from a homogeneous distribution of sites within the

surface. The potential is

(7)

where z is the perpendicular distance of the bead from the surface, and εs controls the strength of the

bead-surface attraction. In our simulations we define εs in terms of the basic energy parameter ε by

defining the dimensionless ratio ε*s = εs/ε. In practice, we concentrated on the values in the range 0.4 ≤

ε*s ≤ 1.

For simulating the bead-spring polymer chains, we used Langevin dynamics in which the system is

coupled to a heat bath, corresponding physically to solvent . In addition to the conservative forces

arising from the interaction potentials described above, each bead will feel random and frictional

forces mimicking the solvent surrounding the bead. Thus the equations of motion for bead i are given

by

(8)

where Γ is the friction coefficient, Wi(t) describes the Brownian forces of the solvent acting on the

bead, and V = ∑i<jVij is the total interaction potential. Wi(t) is represented by Gaussian white noise

satisfying the fluctuation-dissipation theorem48

〈Wi(t)·Wj(t′)〉 = 6kBTmΓδijδ(t−t′) (9)

Simulations were performed in an L × L × H cuboidal box with periodic boundary conditions in all

three directions and the minimum-image convention applied. The box dimension in the z direction

was set to a large, but finite, value of H = 200σ, and the surface was represented by a structureless, L

× L × l solid slab with a thickness l no less than the maximum range of interaction between beads. To

control the surface bead density, L took on values of 80σ through to 180σ, which were always large

enough for polymers in their natural conformations to avoid interacting with their own periodic

images. The simulation conditions mean that the polymers are at finite density within the simulation

cell , and so there is an equilibrium state where the polymers are adsorbed. In principle, the polymers

could adsorb on either face of the slab, but they cannot interact with each other because l is larger than

the interaction range, and H is much larger than typical polymer dimensions (as measured by the

radius of gyration, RG); hence, the two surfaces are essentially isolated from one another. In practice,

initial configurations were prepared by placing the polymers on one face of the slab, and all

Page 7 of 23

subsequent measurements were made for that one surface. Simulated properties are reported here in

reduced units defined in terms of m, ε, and σ. The equations of motion were integrated using the

velocity-Verlet algorithm with a time-step δt = 0.007τ, where is the basic unit of time.

In all cases, the reduced temperature kBT/ε = 1, and the reduced friction coefficient Γτ = 1.

3. Results

We have studied three different situations using Langevin dynamics simulations: (i) the behaviour of

isolated polymers on surfaces with various solvent qualities (with 0 ≤ λ ≤ 1); (ii) the behaviour of

many polymers on surfaces in good-solvent conditions (with λ = 0), corresponding to the experimental

situation before the good solvent-to-bad solvent quench; and (iii) many polymers in bad solvent

conditions (with λ = 1) corresponding to the post-quench situation probed in AFM experiments. We

have studied three different chain lengths (N = 50, 100, and 200 beads), a range of different surface-

energy parameters ε*s = εs/ε, and in the many polymer cases, a variety of surface coverages (to be

defined in Section 3.B). The number of beads in each of the longest chains is of the same order of

magnitude as the number of Kuhn segments in the smallest chains studied in experiments. Such

coarse-grained, bead-spring models are known to reproduce faithfully the semi-quantitative properties

of polymers , and so despite the small lengths of our chains, we anticipate that the various trends seen

in our results will be of relevance to experiments on ultra-thin polymer films after a sudden change in

the solvent quality. Our choices for ε*s are based on achieving a suitable degree of surface adsorption.

We have not attempted the difficult problem of determining this effective interaction parameter from

first principles; this would involve using atomistic representations of the surface, polymer , and

solvent in order to determine the direct and solvent -mediated forces acting between the polymer and

surface.

A. Isolated polymer

The average conformation of a polymer in dilute solution is well understood, and has an isotropic

globular shape defined by a radius of gyration RG, which scales with the number of monomer units N

likeRG Nν where ν is the characteristic Flory exponent equal to 1/3 in a bad solvent , 1/2 in a θ-

solvent, and 3/5 in a good solvent .27

In the proximity of a surface, the number of available polymer

conformations is reduced, leading to a decrease in entropy. Adsorption occurs when the surface-

energy parameter εs exceeds a certain critical value, εcs, signalling that the energetic contribution to the

free-energy from polymer -surface interactions becomes significant.27,49

The accompanying change in

Page 8 of 23

polymer conformation can be interpreted as the order parameter for a type of second-order phase

transition at εs = εcs.

27,49 It is useful to define the dimensionless variable κ

κ = (εs – εcs)/ε

cs (10)

which measures the relative distance from the critical point. We also define the scaling variable y,

y = κN (11)

where is a crossover exponent; this variable appears in the scaling analysis of polymer

adsorption.32,28

Through scaling theory, one can identify four regimes of adsorption depending on the

values of yand κ. (a) For a repulsive wall, y < 0, the chain trivially remains away from the surface. (b)

Near the critical point, y 0, the chain is equally likely to be found at the surface as it is in the

solution. (c) When y ≫ 1 with N large and κ small, adsorption is nonetheless favoured because the

sum of the interactions of the individual beads with the surface outweighs the entropic penalty of the

chain being near the wall. This situation is called the weak coupling limit.32

(d) For κ > 1 the

monomers are strongly attracted to the surface and the chain lies flat. This is called the strong

coupling limit.32

Cases (a)–(c) have been studied extensively in simulations of the particular situation

where each chain is tethered to the surface by one end;25,26

case (d) has been studied in simulations of

free chains.21–23

Earlier simulations using the same type of bead-surface potential as in eqn (7) suggest

that, in reduced units, the critical surface-energy parameter εcs/ε 0.1.

26 The values of ε

*s used in the

current simulations (reported below) correspond to the strong coupling regime.

All isolated-polymer simulations began by placing a linear chain in good-solvent conditions (λ = 0)

close enough to the surface for adsorption to occur. Once the chain had adsorbed, the solvent quality

was adjusted by changing λ to the desired value. The molecule was equilibrated for around 106δt, and

then properties were measured over a production run of 5 × 106δt. Fig. 3(a) and (b) shows the

probability density p(z) of finding beads at a perpendicular distance z from the surface, from

simulations of isolated polymers consisting of N = 100 beads, with various values of ε*s and λ. (The

density profiles are reported in this form—normalized so that ∫∞

0p(z)dz = 1—to aid comparison with

later results for many polymers at finite densities.) In all cases there is either a local minimum or a

point of inflection in p(z) at z 1.2σ, and the position of these features was taken as a distance-based

criterion for assessing whether a particular bead is ‘bound’ to the surface or not. Note that from eqn

(7) the minimum bead-surface potential energy is located at ; this distance

corresponds to the positions of the primary peaks in p(z).

Page 9 of 23

Figure 3. The probability distribution p(z) of finding a bead at a perpendicular distance z from the

surface. All results are for chains with N = 100 beads. (a) Isolated chain with ε*s = 0.4 and 0 ≤ λ ≤ 1.

(b) Isolated chain with λ = 0 (good solvent ) and 0.4 ≤ ε*s ≤ 1.0. (c) Isolated chain compared to many

chains (with densities ρ* = 0.6, 0.8, and 1.0), all with λ = 0 (good solvent ) and ε*s = 1.0. Approximate

ranges of the proximal, central and distal regimes for the system with ρ* = 1.0 are indicated.

In Fig. 4 we show the average bound fraction 〈Φa〉 and the average maximum height 〈h〉 as

functions of λ for different chain lengths and surface-energy parameters ε*s. With N = 50, no

apparently stable adsorption occurs for ε*s < 0.6, while for longer chains with N = 100 and 200,

adsorption occurs when ε*s ≥ 0.4. That is because for ε

*s = 0.4 we are close to the weak coupling

regime, meaning that the individual bead-surface interactions are not sufficient to keep each of them

on the surface. The bound fraction 〈Φa〉 is the fraction of beads within interaction range of the

surface defined using the distance-based criterion z ≤ 1.2σ. For all ε*s, 〈Φa〉 remain insensitive to N.

This is in agreement with theory and simulations, since for κ > 1 the bound fraction scales like Φa

N0.50

It also remains insensitive to solvent quality. There is though only a weak monotonic decrease

with increasing λ; this is due to the polymers bunching up to optimize the attractive bead-bead

interactions, at the cost of bead-surface contacts. Unsurprisingly, for a given λ, increasing ε*s leads to

a greater bound fraction.

← Figure 4. The bound

fraction 〈Φa〉 (top) and the

average maximum height 〈h

〉 (bottom) against solvent

quality λ for isolated linear

chains with, from left to right,

N = 50, 100, and 200 beads,

and with different surface-

interaction parameters ε*s.

Page 10 of 23

In general, the average maximum height 〈h〉 for all systems with ε*s ≥ 0.6 shows a very weak

dependence on λ, there being only a slight hint of an increase as the bad-solvent conditions (λ = 1) are

approached; this is due to the ‘bunching up’ of the beads, to take advantage of their mutual attractive

interactions. But on the whole, the strong bead-surface interactions keep the polymers quite flat on the

surface, with small, short-lived ‘loops’ and ‘tails’ appearing as beads lift off the surface as a result of

thermal fluctuations.

Different behaviour is observed in those systems with N = 100 and 200 beads, and ε*s = 0.4, in which

〈h〉 clearly decreases with increasing λ. This, again, is due to the ‘bunching up’ of the polymer

chain, much like an accordion being compressed. The difference here, though, is that with such a

weak bead-surface interaction parameter, the polymers possess only a small number of contacts with

the surface, leading to the formation of large, long-lived ‘loops’ and ‘tails’ oriented perpendicular to

the surface. When λ is increased, the loops and tails contract, leading to a reduction in the height of

the polymer ; but with a weak bead-surface interaction, this process occurs without the loops

flattening out and forming new contacts with the surface.

B. Many polymers —good solvent

Polymers in good solvent experience purely repulsive mutual interactions. Appropriate simulations

with λ = 0 were initiated by preparing configurations with many ‘curled up’ polymers on a surface,

and equilibrating for around 106δt. For chains of N = 50, 100, and 200 beads, we initially placed 200,

100, and 50 chains on the surface, respectively, leading to the same total number of beads in each

case. Following equilibration, we performed a production run of 2.5 × 106δt. The adsorption is

measured by the equilibrium surface bead density, defined in terms of the number of beads Nads

belonging to those chains with at least one bead-surface contact, defined using the distance-based

criterion z ≤ 1.2σ. Note that Nads is, in general, greater than the number of beads actually bound to the

surface. The reduced surface bead density is ρ* = Nadsσ2/L

2. By placing a fixed number of chains on

surfaces of various sizes, we simulated surface densities up to that corresponding to the critical

overlap concentration. In other words, we approached the semidilute regime within the adsorbed film.

During equilibration runs near the critical overlap concentration, some of the polymers were seen to

desorb as the system approached the steady state.

We first consider the average bound fraction 〈Φa〉 and maximum height 〈h〉, presented in Fig. 5.

Results are shown as functions of the surface bead density ρ* for chains of N = 50, 100, and 200

beads with bead-surface interaction parameters 0.4 ≤ ε*s ≤ 1.0. For comparison, points for isolated

chains are shown at ρ* = 0, the effective density in this case. In all cases, 〈Φa〉 decreases with

Page 11 of 23

increasing ρ*. This may be explained by the entropic penalty associated with a reduced number of

molecular conformations due to crowding; this effect becomes more pronounced as the surface

density is increased. With purely repulsive bead-bead interactions, there is no additional energetic

gain upon adsorption (above the bead-surface interaction) to offset the growing entropic penalty.

Hence, it is more favourable for some beads to lift off the surface to ease crowding. Some additional

observations are that for a given N and ρ*, 〈Φa〉 increases with increasing ε*s; and that for a given

ρ* and ε*s, 〈Φa〉 is essentially independent of N. We note that 〈Φa〉 has been examined in

experiments on linear-polymer films,51

but these were conducted with chemisorbed molecules, as

opposed to the physisorbed molecules considered in this work. Chemisorption reduces sorbate

mobility, and hence reduces the opportunity for reorganization. In addition, molecules can be

irreversibly chemisorbed through fewer surface contacts than those required for physisorption. Both

of these effects lead to relatively low bound fractions, as compared to the results reported here.

Figure 5. The bound fraction 〈Φa〉 (top) and the average maximum height 〈h〉 (bottom) against

surface bead density ρ* for many linear chains with, from left to right, N = 50, 100, and 200 beads, in

good solvent (λ = 0) and with different surface-interaction parameters ε*s. Wherever possible, the

corresponding isolated-polymer results are shown at ρ* = 0.

Examples of the structural differences between weak and strong surface parameter cases are

illustrated in the simulation snapshots shown in Fig. 6. These are from simulations of chains with N =

100 beads. With ε*s = 0.4 and at ρ* = 0.39, the polymers form ‘loops’ and ‘tails’, orientated away

from the surface; with ε*s = 1.0 and at ρ* = 0.59, the polymers are flat on the surface, despite the high

density.

Page 12 of 23

Figure 6. Top views (top) and side views (bottom) from simulations of chains with N = 100 beads in

good solvent (λ = 0). The surface dimensions are 130σ × 130σ. (Left) ε*s = 0.4, ρ* = 0.39, Nads = 6600.

(Right) ε*s = 1.0, ρ* = 0.59, Nads = 10 000. Figures were prepared using Pymol

(http://pymol.sourceforge.net/).

Variations in the average maximum height 〈h〉, shown in Fig. 5, correlate with those in 〈Φa〉. As

the bound fraction decreases, the film height increases, reflecting the build-up of the polymerfilm.

Scaling theory predicts that the height 〈h〉 scales like εsν/(ν–1)

where ν is the characteristic

exponent.27,52,53

For polymers in good solvent , v = 3/5 and so 〈h〉 εs−3/2

. This applies to isolated

polymers , and to many polymers when the surface concentration is much greater than the bulk

concentration. Since in all cases we have an effective bulk density of zero, the scaling law should be

observed. For each system showing significant adsorption under good-solvent conditions, we fitted 〈

h〉 to the function

(12)

and plotted the quantities 〈h〉/h0 on a single graph, as shown in Fig. 7. The results should collapse

on to the curve (ε*s)−3/2

; they are indeed broadly consistent with the predicted scaling. Note that the

critical surface-energy parameter is expected to be unimportant in this analysis, because we are

working in the strong-coupling regime. From the fit shown in Fig. 7, it is clear that the critical value

of ε*s would be small compared to the values used in the simulations. Indeed, attempts to fit critical

parameters led to values of no more than 0.1, in reduced units, but with relative statistical

uncertainties approaching 100%.

Page 13 of 23

Figure 7. Scaling plot of the maximum height 〈h〉 against the surface-interaction parameter ε*s for

polymers in good-solvent conditions (λ = 0). A reduced density of ρ* = 0 corresponds to isolated

polymers . The theoretical prediction27,52,53

is that 〈h〉 εsν/(ν–1)

, which for good-solvent conditions

(v = 3/5) gives 〈h〉 εs−3/2

. h0 is the constant of proportionality from eqn (12).

The effects of the surface-interaction parameter on the conformations of polymers in good-solvent

conditions can be characterized in terms of the radius of gyration RG defined by

(13)

We note that R2G can be decomposed into components perpendicular and parallel to a surface, but the

average value defined above is sufficient for the current purposes. Fig. 8 shows the ratio γ

=R2G(many)/R

2G(isolated), where R

2G (many) is for many polymers made up of N = 200 beads in

good-solvent conditions and at finite density, and R2G (isolated) is the corresponding value for an

isolated polymer on a surface (and therefore at an effective density of ρ* = 0). Results are shown for

systems with various surface-interaction parameters. For a given ε*s, increasing the density causes a

decrease in γ, reflecting a crowding effect due to neighbouring polymers . For a given ρ*, increasing

ε*s causes a flattening of the polymers , and hence an overall reduction in the average dimensions.

Page 14 of 23

Figure 8. R 2 G for many polymers , divided by R

2G for an isolated polymer , against surface bead

density ρ*, for polymers with N = 200 beads in good-solvent conditions and with various surface-

interaction parameters ε*s.

The probability density p(z) for the many-polymer case is shown in Fig. 3(c). Specifically, we show

results for chains consisting of N = 100 beads, with λ = 0, a fixed surface-interaction parameter ε*s=

1.0, and surface bead densities ranging from ρ* = 0 (isolated chain) to ρ* = 1.0. The proximal regime,

identified by de Gennes,32

is dominated by the bead-surface interactions, and in this case covers the

range 0 ≤ z/σ ≤ 2; the two peaks can be interpreted as arising from two ordered layers on the surface.

For an isolated chain, p(z) dies off very rapidly beyond z 2σ. At finite densities, p(z) dies off rapidly

at large distances, roughly corresponding to the distal regime. Under the same conditions, there is an

intermediate, central regime in which p(z) should vary like z−4/3

in good-solvent conditions;27

our

simulation results appear to be consistent with this scaling law. For the system with ρ* = 1.0, the

central regime covers the range 2 ≤ z/σ ≤ 10; approximate ranges of the proximal, central and distal

regimes for this system are indicated in Fig. 3(c). In general, the theoretical scaling predictions should

apply to long chains and to adsorption from semidilute solutions. The experimental31

and simulation

conditions correspond more closely to those of an adsorbed film in contact with a pure solvent ; in

addition, the simulated chains are relatively short. Our results approach the expected z−4/3

scaling with

increasing density, as the conditions near the surface begin to resemble those in a film formed by

adsorption from semidilute solutions. We note that the upper limit of the apparent central regime (

10σ) is comparable to the radius of gyration of an isolated polymer ( Nν) with N = 100 monomers.

We emphasize that this study was not focused on observing the predicted scaling, but it is comforting

Page 15 of 23

that our simulation results are at least consistent with the theoretical predictions;28,29,32

coarse-grained,

off-lattice models of polymers in good solvent appear to form de Gennes' ‘self-similar carpet’.29

C. Many polymers —bad solvent

The final step in the experimental polymer -adsorption procedure being considered here, is the quench

from good-solvent to bad-solvent conditions, corresponding to rinsing with solvent and then drying in

nitrogen /air. In our simulations, we mimic this step by starting simulations from well equilibrated

configurations with λ = 0 (good solvent ) and then instantaneously switching to λ = 1 (bad solvent ).

We then re-equilibrate the system for 2 × 106δt, during which time the system was seen to reach an

apparent steady state. In our experiments, the surfaces were imaged after having been stored at room

temperature ( 125 K above the glass-transition temperature) for periods of several weeks; the

observed cluster structures may well represent the equilibrium, or close-to-equilibrium, state, since the

system has had time and sufficient thermal energy to relax. In our simulations no restrictions exist in

the lateral directions, e.g., surface roughness, thus allowing the polymers slowly to diffuse on the

surface. In earlier simulations of polymer films on surfaces, apparently metastable structures have

been observed for periods of time that might extend towards experimental timescales.54

Simulations

therefore provide valuable insights on the experimental images. Nevertheless, it is easy to imagine

that the true equilibrium state—if accessible—would correspond to a single, large

(hemispherical)polymer droplet;17–19

so the observed behaviour in our simulations might best be

described as metastable. This is because the diffusion rate of an absorbed polymer chain in a bad

solvent is not only lower than that in a good solvent , but also inversely proportional to the number of

beads.20

Thus, within the simulation timescale, clusters and isolated chains may not diffuse

sufficiently far in order to form a putative, single-droplet equilibrium structure. In any case, the

structures we observe are apparently static on the timescales accessed in the simulations.

Fig. 9 shows examples of equilibrated simulation configurations before (λ = 0) and after (λ = 1) the

quench, for systems of polymers each made up from N = 200 beads, with surface-interaction

parameters ε*s = 0.4, and at various densities. In good-solvent conditions, the polymers are in

extended conformations, but in bad-solvent conditions they collapse to form globular clusters to

optimize the attractive bead-bead interactions. At high density (ρ* = 0.81) the quench induces

extensive clustering, resulting in a small number of large clusters . At intermediate density (ρ* =

0.49), a mixture of single chains and large clusters is in evidence, with the single chains in the

minority. At low density (ρ* = 0.30) the single chains are more numerous.

Page 16 of 23

Figure 9. Top views from simulations of chains with N = 200 beads in good-solvent conditions (λ = 0,

top) and bad-solvent conditions (λ = 1, bottom), and with surface-interaction parameter ε*s = 0.4.

(Left) ρ* = 0.81, L = 80σ, Nads = 5200. (Center) ρ* = 0.49, L = 130σ, Nads = 8200. (Right) ρ* = 0.30, L

= 180σ, Nads = 9720. Figures were prepared using Pymol (http://pymol.sourceforge.net/).

In Fig. 10 we show cluster -size histograms for systems of polymers (N = 200 beads per polymer ) in

bad-solvent conditions, with fixed surface bead density ρ* = 0.30, and with various surface-

interaction parameters ε*s. Two polymers were considered clustered if any two beads on different

polymers were within a distance of 1.5σ. Histograms were accumulated from sets of five independent

good solvent-to-bad solvent simulations. With small surface-interaction parameters (ε*s = 0.4 and 0.6)

the distribution shows a monotonic decrease from the peak corresponding to single chains; with larger

parameters (ε*s = 0.8 and 1.0), the distribution is bimodal, with a clear delineation between one-chain

and two-chain species, and larger clusters . The bimodal distribution is to be compared qualitatively to

that found in AFM experiments, Fig. 1(b). A direct, quantitative comparison is not feasible because

we have not considered a specific, coarse-grained molecular model tailored to describe 78.8 kg mol−1

linear PB adsorbed on mica. Another factor that might influence the pattern formation, and that has

not been considered in the simulations, is polydispersity in the lengths of the chains. Nonetheless, we

suggest that the general picture, to be sketched out below, will apply to the real, experimental

situation.

Page 17 of 23

Figure 10. Cluster -size histograms from simulations of polymers with N = 200 beads in bad-solvent

conditions (λ = 1), at a density ρ* = 0.30, and with various surface-interaction parameters. Each

histogram is an average of five independent simulations of the good-solvent to bad-solvent quenching

process. In each case, the total number of adsorbed beads (as defined in section IIIB) isNads = 10 000.

Clearly, the nature of the cluster -size distribution depends on both ρ* and ε*s. From simulations of

200-bead polymers at different densities and with different surface-interaction parameters, we

constructed a ‘phase diagram’ indicating whether the quenched configurations in bad-solvent

conditions showed monotonically decreasing cluster -size distributions, bimodal distributions, or

distributions showing single peaks; it is shown in Fig. 11. In general, low ρ*/low ε*s favours a

monotonically decreasing cluster distribution. Increasing ε*s flattens out the polymers on the surface,

while increasing ρ* slightly leads to more overlaps with neighbouring chains; either change leads to

more pronounced clustering and a bimodal cluster distribution. At high values of ρ* and ε*s, the

clustering is extensive and the cluster -size distribution is strongly peaked (typically at around five

chains per cluster ).

Page 18 of 23

Figure 11. ‘Phase diagram’ in the plane of surface density ρ* and surface-interaction parameter ε*s for

polymers with N = 200 beads in bad-solvent conditions (λ = 1), showing the occurrences of cluster

distributions which are either monotonic decreasing with cluster -size, bimodal, or single peaked.

From these results, we can picture the polymer behaviour during the good solvent-to-bad solvent

quench as follows. At low density and with a low surface-interaction parameter, the polymers are

largely isolated from one another and the probability of forming bead-bead contacts is low because of

the large mean separation and the existence of ‘loops’ and ‘tails’ extending perpendicular to the

surface. The polymer conformations are essentially the same as for an isolated polymer . During the

quench, the majority of polymers simply fold up by themselves; successively smaller proportions of

the molecules form dimers, trimers, etc., leading to a monotonic, rapidly decaying cluster distribution.

At high density and with a high surface-interaction parameter, the polymers are held flat on the

surface and hence the chains form many more contacts with their neighbours. Therefore, during the

quench, polymers aggregate with their neighbours and go on to form large clusters . The cluster

distribution is consequently peaked at a relatively large value. At intermediate values of the density or

the surface-interaction parameter, the polymer conformations are not significantly different from

those of isolated polymers , but there are many more opportunities for forming contacts with

neighbours. These factors favour a mixture of the extremal processes described above, and so give

rise to a bimodal cluster distribution.

Page 19 of 23

4. Conclusions

In this work we have used Langevin dynamics simulations of a coarse-grained, bead-spring model to

gain insight on the adsorption of linear polymers on to a smooth surface. The main experimental

results we set out to understand are AFM images of polymers physisorbed from solution on to mica

surfaces during a process of rapid solvent evaporation . We mimicked this process by switching the

bead-spring model interactions from good-solvent to bad-solvent conditions. Of particular interest

was the experimental observation of a bimodal cluster distribution. We have shown that this feature is

favoured at low-to-moderate polymer concentrations, and over a broad range of polymer -surface

interaction strengths. At high concentrations, a single-peaked distribution is observed; at low

concentrations, and with weak polymer -surface interactions, a monotonically decaying cluster

distribution is obtained. The trends observed in simulations have been rationalized in terms of the

probable numbers of contacts between polymers before quenching from good-solvent to bad-solvent

conditions.

We have measured and rationalized the trends in a variety of other properties including the fraction of

monomer units bound to the surface, the height of the adsorbed polymer film, and the radius of

gyration of an adsorbed polymer chain. An additional, incidental result of this study is the

reproduction of an algebraically decaying density profile within the ‘central regime’, as predicted by

de Gennes using scaling arguments;32

the simulation results are consistent with the prediction that the

monomer density as a function of the perpendicular distance from the surface (z) decays like z−4/3

. The

existence of the central regime has been confirmed experimentally33–36

and in Monte Carlo

simulations of lattice models,37–39

but as far as we are aware, this has not been demonstrated before in

simulations of an off-lattice model.

Future simulation work will be focused on the adsorption and clustering of star polymers on smooth

surfaces. In addition, the kinetics of adsorption and clustering will be explored in detail. For now, we

note that two distinct mechanisms for the self-assembly of adsorbed polymers were identified through

inspecting movies of the simulated quenching process. In the first mechanism, weakly adsorbed

chains first collapse into individual globules, which then slowly diffuse over the surface and coalesce.

The cluster distribution then apparently reaches a steady state on the simulation timescale. This

process was more common with small chains (with N = 50 beads) at low concentrations. The second

mechanism involves the development of contacts between the polymers prior to quenching, i.e., in

good-solvent conditions. Upon quenching, the chains collapse into one another, and form more

extended structures. Occasionally, we observed a chain bridging between two others, and causing all

three to collapse simultaneously. These mechanisms were favoured by longer chains (N = 200 beads)

at high concentrations. A detailed simulation study of the kinetics of the adsorption and clustering

processes is in progress.

Page 20 of 23

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